Gli Algoritmi della Libreria SOFA
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4. Alcuni esempi di calcolo

Per prima cosa vediamo in dettaglio le applicazioni che Patrick Wallace ha riportato, da p.17 a p.27, nel suo ′cookbook′, ovvero il manuale-guida delle routine SOFA, intitolato “Sofa Tools for Earth Attitude”. Possiamo tradurlo come 'Strumenti SOFA per determinare la posizione della Terra nello spazio', in quanto il termine “attitude” sta per `direzione`.

Cominciamo con l'Example 5.2 di p.20-21 del documento suddetto. Riguarda la matrice di precessione-nutazione-bias NPB (equinox based) calcolata tenendo in considerazione le vecchie risoluzioni IAU 1976/1980/1982/1994. Il codice Fortran 90 del file 1.f90, ricco di informazioni riguardanti le impostazioni iniziali del problema e le varie matrici calcolate, è prelevabile da qui, mentre per l'eseguibile DOS (se non si ha tempo per compilarlo) basta scaricare, e poi dezippare, questo file.

L'output si ottiene con il comando: 1 > risu.txt ed è riportato appresso. Si noterà pure come nelle tre matrici sono rispettate le approssimazioni fino alla 15a decimale, ovvero sino alla doppia precisione delle variabili, assunta con il comando REAL(8) di Fortran 90. Naturalmente le stesse coincidono perfettamente con quelle di p.21 del “cookbook”.

   (1.f90) p.20-21
   IAU 1976/1980/1982/1994, equinox based
   ======================================

   Greenwich Sideral Time (GST)=   13.412402380741 deg
                                   0h 53m 38.976571378s
  NPB MATRIX, equinox based
  RNPB (3x3) =   0.999998403176203  -0.001639032970562  -0.000712190961847
                 0.001639000942243   0.999998655799521  -0.000045552846624
                 0.000712264667137   0.000044385492226   0.999999745354454

  Celestial to Terrestrial MATRIX: no polar motion
  RC2TI (3x3)=   0.973104317592263   0.230363826166889  -0.000703332813776
                -0.230363798723540   0.973104570754695   0.000120888299841
                 0.000712264667137   0.000044385492226   0.999999745354454

  Celestial to Terrestrial MATRIX: including polar motion
  RC2IT (3x3)=   0.973104317712771   0.230363826174788  -0.000703163477127
                -0.230363800391874   0.973104570648020   0.000118545116892
                 0.000711560100206   0.000046626645796   0.999999745754057

Proseguiamo adesso con gli altri tre esempi del manuale, inseriti insieme al primo esempio, nei seguenti listati:

Listati
1.f90
2.f90
3.f90
4.f90
Eseguibili Linux
1-Lin.tar.gz
2-Lin.tar.gz
3-Lin.tar.gz
4-Lin.tar.gz
Eseguibili DOS gzippati
1.tar.gz
2.tar.gz
3.tar.gz
4.tar.gz

E questi, in sequenza, sono i risultati:

   (2.f90) p.22-23
   IAU 2000A, CIO based, using classical angles
   ============================================

     Earth Rotation Angle (ERA)=   13.318492966096 deg
                                   0h 53m 16.438311863s
     X=0.000712264729708;  Y=0.000044385250265;  s=-0.002200496 arcsec

  NPB MATRIX, CIO based
  RC2I (3x3) =   0.999999746339445  -0.000000005138721  -0.000712264730182
                -0.000000026475329   0.999999999014975  -0.000044385242666
                 0.000712264729708   0.000044385250265   0.999999745354420

  Celestial to Terrestrial MATRIX: no polar motion
  RC2TI (3x3)=   0.973104317573104   0.230363826247805  -0.000703332818915
                -0.230363798804278   0.973104570735551   0.000120888549767
                 0.000712264729708   0.000044385250265   0.999999745354420

  Celestial to Terrestrial MATRIX: including polar motion
  RC2IT (3x3)=   0.973104317697512   0.230363826239225  -0.000703163482268
                -0.230363800456133   0.973104570632778   0.000118545366806
                 0.000711560162777   0.000046626403835   0.999999745754024

*-----*

   (3.f90) p.23-24
   2000A, equinox based
   ====================
     Date: 2007- 4- 5   Time: 12: 0: 0.00   UT1-UTC=-0.07207 s (from IERS)
       TT [MJD]:   54195.500754444445 = UTC corrected for leap seconds
      UT1 [MJD]:       0.499999165814 = UTC + DeltaT (from IERS)
     DATE [MJD]:   54195.000000000000 = UTC Date at 0 hrs
     Fraction of day:  0.500000000000 = UTC
     --------------------------------

   Greenwich Sideral Time (GST)=   13.412417084674 deg
                                   0h 53m 38.980100322s

  NPB MATRIX, equinox based
  RNPB (3x3) =   0.999998402755640  -0.001639289519579  -0.000712191013215
                 0.001639257491365   0.999998655379006  -0.000045552787477
                 0.000712264729795   0.000044385250265   0.999999745354420

  Celestial to Terrestrial MATRIX: no polar motion
  RC2TI (3x3)=   0.973104317573210   0.230363826247358  -0.000703332818999
                -0.230363798803831   0.973104570735657   0.000120888549787
                 0.000712264729795   0.000044385250265   0.999999745354420

  Celestial to Terrestrial MATRIX: including polar motion
  RC2IT (3x3)=   0.973104317697618   0.230363826238777  -0.000703163482352
                -0.230363800455686   0.973104570632884   0.000118545366826
                 0.000711560162864   0.000046626403835   0.999999745754024

*-----*

   (4.f90) p.26-27
   IAU2006/2000A, CIO based, using X,Y series
   ==========================================
     Date: 2007- 4- 5   Time: 12: 0: 0.00   UT1-UTC=-0.07207 s (from IERS)
       TT [MJD]:   54195.500754444445 = UTC corrected for leap seconds
      UT1 [MJD]:       0.499999165814 = UTC + DeltaT (from IERS)
     DATE [MJD]:   54195.000000000000 = UTC Date at 0 hrs
     Fraction of day:  0.500000000000 = UTC
     --------------------------------

     Earth Rotation Angle (ERA)=   13.318492966096 deg
                                   0h 53m 16.438311863s
     X=0.000712264729525;  Y=0.000044385248875;  s=-0.002200475 arcsec

  NPB MATRIX, CIO based
  RC2I (3x3) =   0.999999746339445  -0.000000005138822  -0.000712264729999
                -0.000000026475226   0.999999999014975  -0.000044385241276
                 0.000712264729525   0.000044385248875   0.999999745354420

  Celestial to Terrestrial MATRIX: no polar motion
  RC2TI (3x3)=   0.973104317573128   0.230363826247706  -0.000703332818416
                -0.230363798804178   0.973104570735574   0.000120888551078
                 0.000712264729525   0.000044385248875   0.999999745354420

  Celestial to Terrestrial MATRIX: including polar motion
  RC2IT (3x3)=   0.973104317697536   0.230363826239126  -0.000703163481769
                -0.230363800456033   0.973104570632801   0.000118545368117
                 0.000711560162594   0.000046626402444   0.999999745754025

A seguire, altri esempi. Eccoli (Aggiornamento del 17.11.2007).

Listati
3set.f90
7set.f90
8set.f90
denver.f90
exa1.f90
exa2.f90
Eseguibili Linux
3set-Lin.tar.gz
7set-Lin.tar.gz
8set-Lin.tar.gz
denver-Lin.tar.gz
exa1-Lin.tar.gz
exa2-Lin.tar.gz
Eseguibili DOS gzippati
3set.tar.gz
7set.tar.gz
8set.tar.gz
denver.tar.gz
exa1.tar.gz
exa2.tar.gz

Ecco pure i tabulati dei risultati degli ultimi 6 esempi.

 (3set.f90)

   IAU2006/2000A, CIO based, using X,Y series [Section A.3 test]
   =============================================================

     Date: 2006- 1-15   Time: 21:24:37.50   UT1-UTC= 0.3341 s
       TT [MJD]:   53750.892855138889 = UTC corrected for leap seconds
      UT1 [MJD]:       0.892104561343 = UTC + DeltaT (from IERS)
     DATE [MJD]:   53750.000000000000 = UTC Date at 0 hrs
     Fraction of day:  0.892100694444 = UTC
     --------------------------------
     Earth Rotation Angle (ERA)=   76.265431053521 deg
                                   5h  5m  3.703452845s
     X(rad)=0.000584859819249;  Y(rad)=0.000041535242468;  s(”)=-0.002571986
     X(”)  = 120.635997299064;  Y(”)=   8.567258740044

  NPB MATRIX, CIO based --> (A.32)
   RC2I =   0.99999982896948092   0.00000000032319160  -0.00058485981976671
           -0.00000002461548598   0.99999999913741178  -0.00004153523517497
            0.00058485981924879   0.00004153524246778   0.99999982810689292

  Celestial to Terrestrial matrix --> (A.33)
  RC2TI =   0.23742421473054331   0.97140604802742336  -0.00017920749985662
           -0.97140588849284648   0.23742427873022337   0.00055827489333994
            0.00058485981924879   0.00004153524246778   0.99999982810689292

*-----*

 (7set.f90)

   IAU2006/2000A, CIO based, using X,Y series [Section A.3 test]
   =============================================================

     Date: 2006- 1-15   Time: 21:24:37.50   UT1-UTC= 0.3341 s
       TT [MJD]:   53750.892855138889 = UTC corrected for leap seconds
      UT1 [MJD]:       0.892104561343 = UTC + DeltaT (from IERS)
     DATE [MJD]:   53750.000000000000 = UTC Date at 0 hrs
     Fraction of day:  0.892100694444 = UTC
     --------------------------------
     Earth Rotation Angle (ERA)=   76.265431053521 deg
                                   5h  5m  3.703452845s
     X(rad)=0.000584859819249;  Y(rad)=0.000041535242468;  s(”)=-0.002571986
 X,Y,Z=   0.00058485981924879   0.00004153524246778   0.99999982810689292rad
 s,ERA=  -0.00000001246933987   1.33108287733667652 rad
 bet,a=   1.33108286486733673   0.50000004297328049 rad
 r11,r12,r13=  0.23742426255380722 -0.97140604787395635 -0.00017920751377925
 r21,r22,r23= -0.97140587680421864  0.23742427935812385  0.00055827488887075

  Celestial to Terrestrial matrix --> (A.33)
  RC2TI =   0.23742421473054331   0.97140604802742336  -0.00017920749985662
           -0.97140588849284648   0.23742427873022337   0.00055827489333994
            0.00058485981924879   0.00004153524246778   0.99999982810689292

*-----*

 (8set.f90)

   IAU2006/2000A, CIO based, using X,Y series [Section A.3 test]
   =============================================================

     Date: 2006- 1-15   Time: 21:24:37.50   UT1-UTC= 0.3341 s
       TT [MJD]:   53750.892855138889 = UTC corrected for leap seconds
      UT1 [MJD]:       0.892104561343 = UTC + DeltaT (from IERS)
     DATE [MJD]:   53750.000000000000 = UTC Date at 0 hrs
     Fraction of day:  0.892100694444 = UTC
     --------------------------------
     Earth Rotation Angle (ERA)=   76.265431053521 deg
                                   5h  5m  3.703452845s
     X(rad)=0.000584859819249;  Y(rad)=0.000041535242468;  s(”)=-0.002571986
 X,Y,Z=   0.00058485981924879   0.00004153524246778   0.99999982810689292 rad
 s,ERA=  -0.00000001246933987   1.33108287733667652 rad
   E,d=   0.07089840047176134   0.00058633286103962 rad
 ang1 = theta-E-s=   1.26018448933425509 rad

  Celestial to Terrestrial matrix --> (A.33)
      R =   0.23742421473054331   0.97140604802742336  -0.00017920749985662
           -0.97140588849284648   0.23742427873022337   0.00055827489333994
            0.00058485981924879   0.00004153524246778   0.99999982810689292

  Matrix R, compact formulation --> R = R3(theta-E-s)*R2(d)*R3(E)
                             being: theta=ERA
                                    E=atan2(X,Y)
                                    d= atan(sqrt(X^2+Y^2/Z^2))
      R =   0.23742421473054320   0.97140604802742336  -0.00017920749985662
           -0.97140588849284637   0.23742427873022329   0.00055827489333994
            0.00058485981924879   0.00004153524246778   0.99999982810689292

*-----*

 (denver.f90)

   IAU2006/2000A, CIO based, using X,Y series
   ==========================================

     Date: 2004- 5-31   Time: 22: 0: 0.00   UT1-UTC=-0.47093 s (from IERS)
       TT [MJD]:   53156.917409537033 = UTC corrected for leap seconds
      UT1 [MJD]:       0.916661216088 = UTC + DeltaT (from IERS)
     DATE [MJD]:   53156.000000000000 = UTC Date at 0 hrs
     Fraction of day:  0.916666666667 = UTC
     --------------------------------
     Given RA -> 23:32:55.171    Given Decl -> 52 16 38.29

     Earth Rotation Angle (ERA)=  219.676330978484 deg
                                  14h 38m 42.319s
  NPB MATRIX, CIO based 
  RC2I(3x3) =  0.99999991755346973  0.00000000726944705 -0.00040607025673205
              -0.00000002029049499  0.99999999948588580 -0.00003206599362056
               0.00040607025629018  0.00003206599921620  0.99999991703935587
  P0 -> initial geocentric versor 
  (x0,y0,z0)=  0.60757413831918361 -0.07212750078667660  0.79098121979979363
  P1 ->   final geocentric versor 
  (x1,y1,z1)=  0.60725289375548264 -0.07215287667632292  0.79122555912516102

       Final RA (= RA1) ->      23h 32m 53.753s
       Final DE (= DE1) ->      52° 18′ 0.683”

*-----*

 (exa1.f90)

   IAU2006/2000A, CIO based, using X,Y series [exa1.f90]
   =====================================================

     Date: 2004- 5-31   Time: 22: 0: 0.00   UT1-UTC=-0.4709 s
       TT [MJD]:   53156.917409537033 = UTC corrected for leap seconds
      UT1 [MJD]:       0.916661216088 = UTC + DeltaT (from IERS)
     DATE [MJD]:   53156.000000000000 = UTC Date at 0 hrs
     Fraction of day:  0.916666666667 = UTC
     --------------------------------
     Earth Rotation Angle (ERA)=  219.676330978484 deg
                                  14h 38m 42.319434836s
     X(rad)=0.000406070256290;  Y(rad)=0.000032065999216;  s(”)=-0.002842323
   Lon(east)= -104.995000 deg;  Lat=  39.742700 deg
   alpha (RA)= 353.229879165 deg
               23h 32m 55.171000s

     Local Hour Angle (HA)=  121.451451813484 deg
                8h  5m 48.348435s

*-----*

 (exa2.f90)

   IAU2006/2000A, CIO based, using X,Y series [exa2.f90]
   =====================================================

     Date: 2004- 5-31   Time: 22: 0: 0.00   UT1-UTC=-0.4709 s
       TT [MJD]:   53156.917409537033 = UTC corrected for leap seconds
      UT1 [MJD]:       0.916661216088 = UTC + DeltaT (from IERS)
     DATE [MJD]:   53156.000000000000 = UTC Date at 0 hrs
     Fraction of day:  0.916666666667 = UTC
     --------------------------------

   Greenwich Sideral Time (GST)=  219.729877844908 deg
                                  14h 38m 55.170682778s

   Earth Rotation Angle   (ERA)=  219.676330978484 deg
                                  14h 38m 42.319434836s
  NPB MATRIX, equinox based
  RNPB (3x3) =   0.999999480862442  -0.000934561717244  -0.000406040936683
                 0.000934548620079   0.999999562783026  -0.000032444310370
                 0.000406071080366   0.000032064828530   0.999999917039059

  Gamma Vector -> (13) of A&A 459 -> NPB 1st row
             =   0.999999480862442  -0.000934561717244  -0.000406040936683
      y Vector -> (14) of A&A 459 -> NPB 2nd row
             =   0.000934548620079   0.999999562783026  -0.000032444310370
  Sigma Vector -> (15) of A&A 459 -> NPB 3rd row
             =   0.999999917553470  -0.000000006510525  -0.000406070256290

   Equation of Origins EO = ERA-GST = -192.768719119 arcsec
                                    =  -12.851247941 s

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